567 research outputs found
Polynomial–exponential equations and Zilber's conjecture
Assuming Schanuel's conjecture, we prove that any polynomial–exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial–exponential equations have only finitely many rational solutions. This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one variable case of Zilber's strong exponential-algebraic closedness conjecture can be reduced to Schanuel's conjecture
Elliptic solutions to a generalized BBM equation
An approach is proposed to obtain some exact explicit solutions in terms of
the Weierstrass' elliptic function to a generalized Benjamin-Bona-Mahony
(BBM) equation. Conditions for periodic and solitary wave like solutions can be
expressed compactly in terms of the invariants of . The approach unifies
recently established ad-hoc methods to a certain extent. Evaluation of a
balancing principle simplifies the application of this approach.Comment: 11 pages, 2 tables, submitted to Phys. Lett.
Sur la possibilité d'une représentation analytique des fonctions dites arbitraires d'une variable réelle
Sur la possibilité d'une représentation analytique des fonctions dites arbitraires d'une variable réelle
An Analytical Construction of the SRB Measures for Baker-type Maps
For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle
and Bowen showed the existence of an invariant measure (SRB measure) weakly
attracting the temporal average of any initial distribution that is absolutely
continuous with respect to the Lebesgue measure. Recently, the SRB measures
were found to be related to the nonequilibrium stationary state distribution
functions for thermostated or open systems. Inspite of the importance of these
SRB measures, it is difficult to handle them analytically because they are
often singular functions. In this article, for three kinds of Baker-type maps,
the SRB measures are analytically constructed with the aid of a functional
equation, which was proposed by de Rham in order to deal with a class of
singular functions. We first briefly review the properties of singular
functions including those of de Rham. Then, the Baker-type maps are described,
one of which is non-conservative but time reversible, the second has a
Cantor-like invariant set, and the third is a model of a simple chemical
reaction . For the second example, the
cases with and without escape are considered. For the last example, we consider
the reaction processes in a closed system and in an open system under a flux
boundary condition. In all cases, we show that the evolution equation of the
distribution functions partially integrated over the unstable direction is very
similar to de Rham's functional equation and, employing this analogy, we
explicitly construct the SRB measures.Comment: 53 pages, 10 figures, to appear in CHAO
How to detect a possible correlation from the information of a sub-system in quantum mechanical systems
A possibility to detect correlations between two quantum mechanical systems
only from the information of a subsystem is investigated. For generic cases, we
prove that there exist correlations between two quantum systems if the
time-derivative of the reduced purity is not zero. Therefore, an
experimentalist can conclude non-zero correlations between his/her system and
some environment if he/she finds the time-derivative of the reduced purity is
not zero. A quantitative estimation of a time-derivative of the reduced purity
with respect to correlations is also given. This clarifies the role of
correlations in the mechanism of decoherence in open quantum systems.Comment: 7 pages, 1 figur
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